Given that `veca,vecc,vecd` are coplanar the value of `(veca + vecd).{veca xx [vecb xx (vecc xx veca)]}` is:

Options

A. `-2`
B. 0
C. 2
D. None of these

Answer

To solve the problem, we need to evaluate the expression \((\vec{a} + \vec{d}) \cdot (\vec{a} \times [\vec{b} \times (\vec{c} \times \vec{a})])\) given that the vectors \(\vec{a}\), \(\vec{c}\), and \(\vec{d}\) are coplanar.

### Step-by-Step Solution:

1. **Understanding Coplanarity**:
Since vectors \(\vec{a}\), \(\vec{c}\), and \(\vec{d}\) are coplanar, we know that the scalar triple product of these vectors is zero:
\[ \vec{a} \cdot (\vec{c} \times \vec{d}) = 0. \]

2. **Using Vector Triple Product Identity**:
We can simplify the expression \(\vec{b} \times (\vec{c} \times \vec{a})\) using the vector triple product identity:
\[ \vec{b} \times (\vec{c} \times \vec{a}) = (\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}. \]

3. **Substituting Back**:
Substitute this result back into the original expression:
\[ (\vec{a} + \vec{d}) \cdot (\vec{a} \times [(\vec{b} \cdot \vec{a}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{a}]). \]

4. **Distributing the Dot Product**:
Now we distribute the dot product:
\[ = (\vec{a} + \vec{d}) \cdot \left((\vec{b} \cdot \vec{a}) (\vec{a} \times \vec{c}) - (\vec{b} \cdot \vec{c}) (\vec{a} \times \vec{a})\right). \]

5. **Evaluating Cross Product**:
We know that \(\vec{a} \times \vec{a} = \vec{0}\) (the zero vector), so the second term becomes zero:
\[ = (\vec{a} + \vec{d}) \cdot \left((\vec{b} \cdot \vec{a}) (\vec{a} \times \vec{c})\right). \]

6. **Factoring Out Scalar**:
Since \((\vec{b} \cdot \vec{a})\) is a scalar, we can factor it out:
\[ = (\vec{b} \cdot \vec{a}) \left((\vec{a} + \vec{d}) \cdot (\vec{a} \times \vec{c})\right). \]

7. **Evaluating the Dot Product**:
Now, we evaluate \((\vec{a} + \vec{d}) \cdot (\vec{a} \times \vec{c})\). Since \(\vec{a} \times \vec{c}\) is perpendicular to both \(\vec{a}\) and \(\vec{c}\), the dot product with \(\vec{a}\) is zero:
\[ (\vec{a} + \vec{d}) \cdot (\vec{a} \times \vec{c}) = \vec{a} \cdot (\vec{a} \times \vec{c}) + \vec{d} \cdot (\vec{a} \times \vec{c}) = 0 + \vec{d} \cdot (\vec{a} \times \vec{c}). \]

8. **Final Result**:
Since \(\vec{a}, \vec{c}, \vec{d}\) are coplanar, \(\vec{d} \cdot (\vec{a} \times \vec{c}) = 0\). Therefore:
\[ = (\vec{b} \cdot \vec{a}) \cdot 0 = 0. \]

Thus, the final answer is:
\[ \boxed{0}. \]